关于用巴比伦法（牛顿-拉夫逊法）计算斜边的最优初始猜测值的问询

Great question! Your initial guess of |a| + |b| is technically valid, but it’s way larger than the actual hypotenuse c—we can do much better by leveraging the geometry of right triangles and properties of square roots. A good initial guess cuts down the number of Babylonian iterations drastically (sometimes even to one!), so let’s break down the best options:

Top Initial Guess Candidates #

1. (|a| + |b|) / √2 (or 0.7071 * (|a| + |b|) for approximation) #

This is my go-to for general cases. Here’s why it works:

• When |a| = |b|, this guess is exactly equal to c (since c = a√2 = (a+a)/√2).
• For uneven sides, it’s still extremely close. Take your classic 3-4-5 triangle: (3+4)/1.4142 ≈ 4.949, which is only 0.051 away from the true hypotenuse 5. Run one Babylonian iteration on this:

  (4.949 + 25/4.949)/2 ≈ (4.949 + 5.051)/2 = 5
  

  You get the exact value in one step!

2. max(|a|, |b|) + (min(|a|, |b|))² / (2 * max(|a|, |b|)) #

This is a Taylor series approximation tailored for when one side is significantly longer than the other (e.g., |a| > 2|b|). Let’s say a=100, b=10:

• True c = sqrt(10000 + 100) = sqrt(10100) ≈ 100.498756
• This guess gives: 100 + (10²)/(2*100) = 100 + 100/200 = 100.5
  That’s practically identical to the true value—you’d need zero or one iteration to converge perfectly.

3. max(|a|, |b|) * 1.2 (quick rough guess) #

If you don’t want to compute divisions or squares for the initial guess, this simple scaling works well. It’s bounded between max(a,b) (the lower limit of c) and max(a,b)*√2 ≈ 1.414 (the upper limit when a=b). For most cases, it’s way closer than |a| + |b|.

Why These Are Better Than |a| + |b| #

Your initial guess is an upper bound, but it’s a very loose one. For the 3-4-5 triangle, |a|+|b|=7 is 40% larger than the true c=5—that means you’d need 2-3 iterations to get to a precise value, whereas the options above often get there in 0 or 1 steps.

The Babylonian method has quadratic convergence, so starting closer to the true value exponentially reduces the number of iterations needed.

内容的提问来源于stack exchange，提问作者Wingblade